Given a set H of graphs, let fH⋆:N>0→N>0 be the optimal χ-binding function of
the class of H-free graphs, that is,
fH⋆(ω)=max{χ(G):G is H-free, ω(G)=ω}. In this paper, we combine the
two decomposition methods by homogeneous sets and clique-separators in order to
determine optimal χ-binding functions for subclasses of P5-free graphs
and of (C5,C7,…)-free graphs. In particular, we prove the following
for each ω≥1:
(i) f{P5,banner}⋆(ω)=f3K1⋆(ω)∈Θ(ω2/log(ω)),
(ii) $\
f_{\{P_5,co-banner\}}^\star(\omega)=f^\star_{\{2K_2\}}(\omega)\in\mathcal{O}(\omega^2),(iii)\
f_{\{C_5,C_7,\ldots,banner\}}^\star(\omega)=f^\star_{\{C_5,3K_1\}}(\omega)\notin
\mathcal{O}(\omega),and(iv)\ f_{\{P_5,C_4\}}^\star(\omega)=\lceil(5\omega-1)/4\rceil.Wealsocharacterise,foreachofourconsideredgraphclasses,allgraphsGwith\chi(G)>\chi(G-u)foreachu\in V(G).Fromthesestructuralresults,wecanproveReed′sconjecture−−relatingchromaticnumber,cliquenumber,andmaximumdegreeofagraph−−for(P_5,banner)$-free graphs